Operations Research Homework #VI
Due date: Jan. 2, 2006 1. Use Gomory cutting plane method to solve the following integer linear programming:
max 3x1−x2
s.t. 3x1−2x2 ≤3
−5x1−4x2 ≤ −10 2x1+ x2 ≤5 x1, x2 ≥0, x1, x2 integers.
2. A company needs to ship a product from m locations to n destinations. Suppose that ai
units of the product are available at the ith origin (i = 1, 2, · · · , m), bj units are required at jth destination (j = 1, 2, · · · , n). Assume that the total amount of available units at all origins equals the total amount required at all destinations. The cost of shipping one unit of product from origin i to destination j is cij and you are asked to minimize the transportation cost. Given that i = 3, j = 4, a1 = 3, a2 = 3, a3 = 4; b1 = 2, b2 = 3, b3 = 2 and b4 = 3 with the cost matrix
C =
7 2 −2 8 19 5 −2 12
5 8 −9 3
Please establish the network and use the Network Simplex method to obtain an optimal solution.
3. Consider the following integer linear programming:
max 2x1+ 3x2
s.t. 195x1 + 273x2≤ 1365 4x1+ 40x2 ≤140 x1 ≤4
x1, x2 ≥0, x1, x2 integers.
Use Branch and Bound method to find an optimal solution.
